Convergence of the generalized volume averaging method on a convection-diffusion problem:: A spectral perspective

This paper proposes a thorough investigation of the convergence of the volume averaging method described by Whitaker [The Method of Volume Averaging, Kluwer Academic, Norwell, MA, 1999] as applied to convection-diffusion problems inside a cylinder. A spectral description of volume averaging brings to the fore new perspectives about the mathematical analysis of those approximations. This spectral point of view is complementary with the Lyapunov-Schmidt reduction technique and provides a precise framework for investigating convergence. It is shown for convection-diffusion inside a cylinder that the spectral convergence of the volume averaged description depends on the chosen averaging operator, as well as on the boundary conditions. A remarkable result states that only part of the eigenmodes among the infinite discrete spectrum of the full solution can be captured by averaging methods. This leads to a general convergence theorem ( which was already examined with the use of the center manifold theorem [ G. N. Mercer and A. J. Roberts, SIAM J. Appl. Math., 50 ( 1990), pp. 1547-1565] and investigated with Lyapunov-Schmidt reduction techniques [ S. Chakraborty and V. Balakotaiah, Chem. Engrg. Sci., 57 ( 2002), pp. 2545-2564] in similar contexts). Moreover, a necessary and sufficient condition for an eigenvalue to be captured is given. We then investigate specific averaging operators, the convergence of which is found to be exponential.